We can use binomial distribution using p=q=0.5, which is a 50% probability of getting an answer right or of getting it wrong. We use the expansion:
(p+q)¹⁰=1=p¹⁰+10p⁹q+45p⁸q²+120p⁷q³+210p⁶q⁴+...
From this expansion we can see that 210p⁶q⁴ is the probability for exactly 6 right answers out of 10.
Since p=q, this comes to 210(0.5)¹⁰=0.2051 or 20.51%. (The same probability applies to getting exactly 6 wrong answers (exactly 4 right answers).)